The Turkey international added his second six minutes from time with a diving header to give Everton victory over 10-man Stoke, who lost Charlie Adam to a first-half red card for a two-footed lunge on Wayne Rooney.
"Under the circumstances we are extremely pleased," Allardyce told BBC Sport. "Considering the elements both teams competed with today they've given everything - even Stoke after they went down to 10 men.
"It looked a bit iffy for us again but Cenk popped up with a super goal and that's why we bought him. It's paid dividends for us and it puts us on the 40-point mark, and even though mathematically we are not safe I don't think anyone can catch us now and we can start to prepare for next season.
"I think the quality of the finish caught everyone out, including the goalkeeper, but it was very difficult today to put any quality together because of the wind and snow. Wayne Rooney kept us going in midfield and kept us prodding but breaking down 10-men can be difficult.
"There weren't that many chances and any chance can come in today's weather - anything could happen, likely a mistake in these conditions and luckily we were the ones that capitalised on that today."
Everton return to action on March 31 with a game against Manchester City at Goodison Park.